Filip Saidak's Proof 
(From the Prime Pages' list of proofs)


Home
Search Site

Largest
The 5000
Top 20
Finding
How Many?
Mersenne

Glossary
Prime Curios!
Prime Lists

FAQ
e-mail list
Titans

Submit primes

Euclid may have been the first to give a proof that there are infintely many primes.  Below we give another proof by Filip Saidak [Saidak2005], similar to Goldbach's argument, but in a way even simpler.
Theorem.
There are infinitely many primes.

Proof.
Let n > 1 be a positive integer.  Since n and n+1 are consecutive integers, they must be coprime, and hence the number N2 = n(n + 1) must have at least two different prime factors.  Similarly, since the integers n(n+1) and n(n+1)+1 are consecutive, and therefore coprime, the number N3 = n(n + 1)[n(n + 1) + 1] must have at least 3 different prime factors.  This can be continued indefinitely.
The Prime Pages
Another prime page by Chris K. Caldwell <caldwell@utm.edu>